A348426 Numbers k for which sigma(k) = k + k'', where k'' is the second derivative of k (A068346).

1, 161, 209, 221, 4265, 12690, 15941, 22217, 24041, 25637, 30377, 38117, 39077, 48617, 49097, 55877, 68441, 73817, 76457, 80357, 88457, 95237, 98117, 99941, 105641, 110057, 115397, 122537, 130217, 131141, 136517, 143237, 147941, 148697, 152357, 154457, 159077

Offset: 1

Author

Marius A. Burtea, Oct 18 2021

Keywords

• nonn

Comments

If p and q are different prime numbers and p + q is in A007850 (Giuga numbers) then m = p*q is a term because sigma(m) = sigma(p*q) = p*q + p + q + 1 and m + m'' = p*q + (p + q)' = p*q + p + q + 1 and sigma(m) = m + m''.

Examples

			sigma(1) = 1 and 1 + 1'' = 1 so 1 is a term.
sigma(161) = 1 + 7 + 23 + 161 = 192 and 161 + 161'' = 161 + 30' = 161 + 31 = 192 so 161 is a term.
sigma(12690) = sigma(2*3^3*5*47) = 34560 and 12690 + 12690'' = 12690 + A068346(12690) = 12690 + 21870 = 34560 so 12690 is a term.
		

Crossrefs

Cf. A000203, A003415, A068346, A007850.

Programs

Magma

f:=func; [n:n in [1..160000]| DivisorSigma(1,n) eq n+Floor(f(Floor(f(n))))];

Mathematica

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[160000], DivisorSigma[1, #] == # + d[d[#]] &] (* Amiram Eldar, Oct 18 2021 *)

PARI

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = sigma(k) == k+ad(ad(k)); \\ Michel Marcus, Oct 18 2021